4. One Dimensional Heat Conduction Equation - For Various Coordinates

Rectangular Coordinates:

Here, the area \(A\) does not vary with \(x\). Hence, Eqn.(2) becomes, \[\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) + \dot{g} = \rho C_P\frac{\partial T(x,t)}{\partial t}\] 

Cylindrical Coordinates:

Here, \(x=r\). Area, \(A=2\pi rL\). i.e., \(A\propto r\). Hence, Eqn.(2) becomes, \[\frac{1}{r}\frac{\partial}{\partial r}\left(rk\frac{\partial T}{\partial r}\right) + \dot{g} = \rho C_P\frac{\partial T(r,t)}{\partial t}\] 

Spherical Coordinates:

Here too, \(x=r\). Area, \(A=4\pi r^2\). i.e., \(A\propto r^2\). Hence, Eqn.(2) becomes, \[\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2k\frac{\partial T}{\partial r}\right) + \dot{g} = \rho C_P\frac{\partial T(r,t)}{\partial t}\]


Compact Equation

The above equations (rectangular / cylindrical / spherical coordinates) can be written in a compact form, as below: \[{\frac{1}{r^n} \frac{\partial}{\partial r}\left(r^nk\frac{\partial T}{\partial r}\right) + \dot{g} = \rho C_P\frac{\partial T}{\partial t} \tag*{(3)} }\] where \[n = \left\{\begin{array}{ll} 0 & \text{for rectangular coordinates} \\ 1 & \text{for cylindrical coordinates} \\ 2 & \text{for spherical coordinates} \end{array} \right.\] And, in the rectangular coordinates system, it is customary to use the variable \(x\) in place of \(r\).


Special Cases

For constant thermal conductivity \(k\), Eqn.(3) reduces to, \[\frac{1}{r^n} \frac{\partial}{\partial r}\left(r^n\frac{\partial T}{\partial r}\right) + \frac{1}{k}\dot{g} = \frac{1}{\alpha}\frac{\partial T}{\partial t}\] where \[\alpha = \frac{k}{\rho C_P} = \text{thermal diffusivity of material, m$^2$/s}\]

For steady state heat conduction with energy sources within the medium, Eqn.(3) reduces to, \[\frac{1}{r^n} \frac{d}{d r}\left(r^nk\frac{d T}{d r}\right) + \dot{g} = 0\] and for the case of conduction with constant \(k\), \[\frac{1}{r^n} \frac{d}{d r}\left(r^n\frac{d T}{d r}\right) + \frac{1}{k}\dot{g} = 0\]

For steady state conduction, with no energy sources, and for constant \(k\), \[\frac{d}{d r}\left(r^n\frac{d T}{\partial r}\right) = 0\]